Spherical Harmonics

Students should be able to:

1. (Sphere)

   Consider the normalized function:

   $$f(\theta,\phi)= \begin{cases} N\left({\pi^2\over 4}-\theta^2\right)&0<\theta<\frac{\pi}{2}\\ 0&{\pi\over 2}<\theta<\pi \end{cases} $$

   where

   $$N=\frac{1}{\sqrt{\frac{\pi^5}{8} +2\pi^3-24\pi^2+48\pi}}$$

   1. Find the $\left|\ell,m\right\rangle=\left|0,0\right\rangle$, $\left|1,-1\right\rangle$, $\left|1,0\right\rangle$, and $\left|1,1\right\rangle$ terms in a spherical harmonics expansion of $f(\theta,\phi)$.

   2. If a quantum particle, confined to the surface of a sphere, is in the state above, what is the probability that a measurement of the square of the total angular momentum will yield $2\hbar^2$? $4\hbar^2$?

   3. If a quantum particle, confined to the surface of a sphere, is in the state above, what is the probability that the particle can be found in the region $0<\theta<{\pi\over 6}$ and $0<\phi<{\pi\over 6}$? Repeat the question for the region ${5\pi\over 6}<\theta<{\pi}$ and $0<\phi<{\pi\over 6}$. Plot your approximation from part (a) above and check to see if your answers seem reasonable.